Methods of critical value reduction for type-2 fuzzy variables and their applications

A type-2 fuzzy variable is a map from a fuzzy possibility space to the real number space; it is an appropriate tool for describing type-2 fuzziness. This paper first presents three kinds of critical values (CVs) for a regular fuzzy variable (RFV), and proposes three novel methods of reduction for a...

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Published in:	Journal of computational and applied mathematics Vol. 235; no. 5; pp. 1454 - 1481
Main Authors:	Qin, Rui, Liu, Yan-Kui, Liu, Zhi-Qiang
Format:	Journal Article
Language:	English
Published:	Kidlington Elsevier B.V 2011 Elsevier
Subjects:	Calculus of variations and optimal control Critical value Data envelopment analysis Exact sciences and technology Fuzzy Fuzzy logic Fuzzy possibility theory Fuzzy set theory Mathematical analysis Mathematical models Mathematics Numerical analysis Numerical analysis. Scientific computation Numerical methods in mathematical programming, optimization and calculus of variations Numerical methods in optimization and calculus of variations Parametric programming Programming Reduction Reduction methods Sciences and techniques of general use Type-2 fuzzy variable Critical value Reduction methods Type-2 fuzzy variable Fuzzy possibility theory Parametric programming Data envelopment analysis Data analysis Optimization method Linear programming Linear model Reduction method Numerical analysis Applied mathematics Variational calculus Mathematical programming
ISSN:	0377-0427, 1879-1778
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Abstract	A type-2 fuzzy variable is a map from a fuzzy possibility space to the real number space; it is an appropriate tool for describing type-2 fuzziness. This paper first presents three kinds of critical values (CVs) for a regular fuzzy variable (RFV), and proposes three novel methods of reduction for a type-2 fuzzy variable. Secondly, this paper applies the reduction methods to data envelopment analysis (DEA) models with type-2 fuzzy inputs and outputs, and develops a new class of generalized credibility DEA models. According to the properties of generalized credibility, when the inputs and outputs are mutually independent type-2 triangular fuzzy variables, we can turn the proposed fuzzy DEA model into its equivalent parametric programming problem, in which the parameters can be used to characterize the degree of uncertainty about type-2 fuzziness. For any given parameters, the parametric programming model becomes a linear programming one that can be solved using standard optimization solvers. Finally, one numerical example is provided to illustrate the modeling idea and the efficiency of the proposed DEA model.
AbstractList	A type-2 fuzzy variable is a map from a fuzzy possibility space to the real number space; it is an appropriate tool for describing type-2 fuzziness. This paper first presents three kinds of critical values (CVs) for a regular fuzzy variable (RFV), and proposes three novel methods of reduction for a type-2 fuzzy variable. Secondly, this paper applies the reduction methods to data envelopment analysis (DEA) models with type-2 fuzzy inputs and outputs, and develops a new class of generalized credibility DEA models. According to the properties of generalized credibility, when the inputs and outputs are mutually independent type-2 triangular fuzzy variables, we can turn the proposed fuzzy DEA model into its equivalent parametric programming problem, in which the parameters can be used to characterize the degree of uncertainty about type-2 fuzziness. For any given parameters, the parametric programming model becomes a linear programming one that can be solved using standard optimization solvers. Finally, one numerical example is provided to illustrate the modeling idea and the efficiency of the proposed DEA model.
Author	Liu, Yan-Kui Qin, Rui Liu, Zhi-Qiang
Author_xml	– sequence: 1 givenname: Rui surname: Qin fullname: Qin, Rui email: qinrui07@163.com organization: College of Mathematics & Computer Science, Hebei University, Baoding 071002, Hebei, China – sequence: 2 givenname: Yan-Kui surname: Liu fullname: Liu, Yan-Kui email: yliu@hbu.edu.cn, liuyankui@tsinghua.org.cn organization: College of Mathematics & Computer Science, Hebei University, Baoding 071002, Hebei, China – sequence: 3 givenname: Zhi-Qiang surname: Liu fullname: Liu, Zhi-Qiang email: zq.liu@cityu.edu.hk organization: School of Creative Media, City University of Hong Kong, Hong Kong, China
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Keywords	Critical value Reduction methods Type-2 fuzzy variable Fuzzy possibility theory Parametric programming Data envelopment analysis Data analysis Optimization method Linear programming Linear model Reduction method Numerical analysis Applied mathematics Variational calculus Mathematical programming
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Snippet	A type-2 fuzzy variable is a map from a fuzzy possibility space to the real number space; it is an appropriate tool for describing type-2 fuzziness. This paper...
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SubjectTerms	Calculus of variations and optimal control Critical value Data envelopment analysis Exact sciences and technology Fuzzy Fuzzy logic Fuzzy possibility theory Fuzzy set theory Mathematical analysis Mathematical models Mathematics Numerical analysis Numerical analysis. Scientific computation Numerical methods in mathematical programming, optimization and calculus of variations Numerical methods in optimization and calculus of variations Parametric programming Programming Reduction Reduction methods Sciences and techniques of general use Type-2 fuzzy variable
Title	Methods of critical value reduction for type-2 fuzzy variables and their applications
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